Gettings bounds for seminorms from bound of absolute value

On a compact domain $$\Omega \subseteq \mathbb{R}^d$$, we have a function $$u(x) \in C^{\infty}(\Omega)$$ with an approximation $$u_h(x)$$ with the following properties:

$$|u(x) - u_h (x)| \leq C h^{m+1} |u(x)|_{C^{m+1}(\Omega)},$$

where the seminorm on the right is defined as $$|u(x)|_{C^{m+1}(\Omega)} = \text{max}_{|\alpha|=m+1} \|D^{\alpha} u\|_{L^{\infty}}$$, with the derivative being defined using the multi-index notation:

$$D^{\alpha} f = \frac{\partial^{|\alpha|}}{\partial x_1^{\alpha_1} \cdots\partial x_n^{\alpha_n}}, \quad \alpha = (\alpha_1, \cdots, \alpha_n),$$

for $$|\alpha| = \alpha_1 + \alpha_2 + \cdots \alpha_n| \leq k$$, $$x \in \mathbb{R}^d$$, $$f \in C^k (\Omega)$$. I want to use that approximation to derive similar bound for the seminorm $$|v|^2_{k, \Omega}$$ defined as the sum of all the squares of the $$L^2$$ norms of all the derivatives of order $$k$$. So in 2D, this seminorm would look like: $$\|v\|^2_{0, \Omega} = |v|^2_{0, \Omega} = \int_{\Omega} v^2 d\Omega \\ |v|^2_{1, \Omega} = \left| \frac{\partial v}{\partial x_1} \right|^2_{0, \Omega} + \left| \frac{\partial v}{\partial x_2} \right|^2_{0, \Omega} \\ |v|^2_{2, \Omega} = \left| \frac{\partial^2 v}{\partial x_1^2} \right|^2_{0, \Omega} + \left| \frac{\partial^2 v}{\partial x_1 \partial x_2} \right|^2_{0, \Omega} + \left| \frac{\partial^2 v}{\partial^2 x_2} \right|^2_{0, \Omega}$$ To be more precise, I'm trying to arrive at some bound for $$|u(x) - u_h(x)|_{s, \Omega}$$, with $$0 \leq s < m$$. Any suggestions how I could achieve that?

• Hi from Vietnam Mar 22 at 6:46
• Use a suitable bump function. That is pick $\phi\in C^\infty(\Omega)$ such that $\phi=1$ on the interior of $\Omega$ and $\phi=0$ on $\partial\Omega$. Then use repeated applications of integration by parts to put all the derivatives on $\phi$ and then use the assumed bound on $|u-u_h(x)|$. Mar 30 at 16:23